On commuting automorphisms and central automorphisms of finite 2-groups of almost maximal class

Let G be a finite 2-group. In our recent papers, we proved that in a finite 2-group of almost maximal class, the set of all commuting automorphisms, A(G) = {alpha is an element of Aut(G) : x alpha(x) = alpha(x)x for all x is an element of G) is equal to the group of all central automorphisms, Aut(c)(G), except only for five ones. Also, we determined the structure of Aut(c)(G) and A(G) for these five groups. Using these results, in this paper, we find the structure of A(G) = Aut(c)(G) for the remaining 2-groups of almost maximal class. Also, we prove the following results: (1) We characterise the upper central series of these groups. (2) We find the necessary and sufficient conditions on 2-groups of almost maximal class in order that A(Inn(G)) to be equal to Aut(c)(Inn(G)) or Aut(c)(G) to be equal to Z (Inn(G)), that is, Aut(c)(G), is as small as possible. (3) Also, we show that if G is a group of almost maximal class and order 2(n), n >= 7 in which 2 <= d(G') <= 3, then Aut(G) is an A-group.